are shown,
for each fixed \(L = 0,\ldots ,K_{\max },\ L \ne K\) , where \(a_n\)  is a positive sequence. However while these immediately give consistency when \(K\)  and \(K_{\max }\)  are finite, it is not clear to us how the arguments may be extended directly to allow for diverging \(K\)  and \(K_{\max }\)  as this would require delicate control of the rates at which the probabilities above approach 1.
Several of the lemmas below, which we require in our proof of Lemma REF , are therefore uniform versions of Lemmas 1–5 in [1]}.
